Oscillating Random Walks with Two Levels of Switching

We consider a Markov random walk $X_n$, $n\ge 0$, generated by the sums of independent random variables. Each successive jump of the random walk is distributed in accord with one of three laws in dependence on the location of a walking particle: within some interval $[a,b]$, to the left of the point $a$, or to the right of the point $b$. Using factorization methods, we obtain some representations for the double Laplace–Stieltjes transforms (in time and spatial variables) of the distribution of $X_n$ and find the transforms of the stationary distribution of a chain.